Lorentz violation induced vacuum birefringence and its astrophysical consequences

In the electromagnetism of loop quantum gravity, two helicities of a photon have different phase velocities and group velocities, termed as "vacuum birefringence". Two novel phenomenons, "peak doubling" and "de-polarization", are expected to appear for a linearly polarized light from astrophysical sources. We show that the criteria to observe these two phenomenons are the same. Further, from recently observed $\gamma$-ray polarization from Cygnus X-1, we obtain an upper limit $\sim 8.7\times10^{-12}$ for Lorentz-violating parameter $\chi$, which is the most firm constraint from well-known systems. We also suggest to analyze possible existence of "peak doubling" through Fermi LAT GRBs.